Fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems, and baseline data with students in a traditional mathematics curricula were used as a comparison. The students in the reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. »Jump to indexing (document details)
==Full Text== (4519 words)
Copyright School Science and Mathematics Association, Incorporated Oct 1996
[Headnote]
Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study,fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with "invented" algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.
When faced with a problem involving mathematical calculations, various solution methods are possible, including the use of paper-and-pencil algorithms, a calculator, or a mental solution. Although schools have generally emphasized standard written algorithms in both instruction and testing, the calculator has received an increasing acceptance as a regular tool from the primary grades onward (e.g., Wheatley & Shumway, 1992). In contrast, mental arithmetic is underrepresented in instruction and is rarely included on high-stakes tests. This is surprising, given that mental calculations and estimations are more common in everyday life than are the paper-and-pencil algorithms practiced in school (Hope, 1986). Current mathematics reform initiatives have recognized the need to align school and everyday uses of mathematics by including mental arithmetic and estimation as regular components in the curriculum (NCTM, 1989).
Besides better preparing students for the mathematics encountered outside of school, experiences with mental computation should help to develop flexible mathematical reasoning, estimation skills, alternative ways of expressing numbers, and the relationship between operations. Rather than focusing simply on the product, (i.e., the correct answer), mental computation allows greater opportunity to view mathematical problem solving as a process in a domain of interconnected knowledge sometimes called number sense (Sowder, 1992). For example, when adding 98 + 17, people with a good number sense may simplify the problem to (98 + 2) + IS or 100 + 20 - 5. They may also note that the sum will be about, but not quite, 120. Experiences with mental arithmetic also help students to recognize when a problem lends itself to a mental calculation and when it is better to use a calculator or pencil, metacognitive skills such as these are important in mathematical problem solving (Schoenfeld, 1992).
Perhaps most importantly, use of mental computation is motivating for many students. For example, in a recent survey (Carroll & Porter, 1994), fourth graders were asked to compare the mathematical knowledge of those who used a mental, paper-pencil, or calculator solution. Nearly all fourth-grade students surveyed felt that the student who used mental computation were better at mathematics.
Good mental calculators use procedures not necessarily learned in school (Dowker, 1992; Hope & Sherrill, 1987; Reys, Bestgen, Rybolt, & Wyatt, 1982) but which build upon number and operation relationships in an efficient manner. For example, good mental calculators often add from left-to-right, recombining numbers as they go along, thus reducing the load on memory. However, without frequent opportunities to develop mental solutions or modeling by the teacher or other students, many students do not develop these efficient techniques. Instead, they attempt to apply standard pencil-and-paper algorithms mentally (Hope & Sherrill, 1987). Due to the strain on short-term memory, these right-to-left calculations with "carrying" and "borrowing" are error-prone when used mentally.
A recent study of by Reys, Reys, and Hope (1993) attempted to set benchmarks in mental computation for comparative purposes. Results on this whole-class test, which tested classes using traditional mathematics curricula, highlighted weaknesses of students in mental computation. For example, only about one-third of fifth graders could mentally add 47 and 29 or could solve a story problem which required subtracting 65 from 100. Few problems were solved by more than half of the students. A preference survey indicated that students chose pencil-and-paper as their preferred method on most problems, even those that lent themselves to mental calculation. The calculator was the least preferred method on all items.
While traditional mathematics curricula emphasize practice of paper-and-pencil algorithms, a number of new mathematics curricula have attempted to incorporate ideas recommended by the current reform movement (NCTM, 1989, 1991; NRC, 1989). In these curricula, the development of estimation skills and number sense of answers is given more emphasis than paper-and-pencil computation. Primary students are often not taught specific algorithms but instead are encouraged to develop and share their own methods. Often these "invented" algorithms are mental and reflect methods used by expert mental calculators, e.g., adding left to right or decomposing numbers (Carroll & Balfanz, 1995; Porter & Carroll, 1995). One such curriculum in wide use is the University of Chicago School Mathematics Project elementary program, Everyday Mathematics. In addition to inventing their own algorithms, students in this curriculum are encouraged to share multiple solutions during group work and whole-class discussion. Manipulatives and calculators are used regularly during problem solving, and computation problems are nearly always presented in an applied situation or in a game. Mental arithmetic is emphasized in the curriculum, both implicitly by encouraging invented algorithms, and explicitly by encouraging the sharing of alternative strategies and short mental arithmetic exercises.
For example, in a typical lesson fourth-grade UCSMP lesson observed, students used with a road map showing various routes and mileage between cities. Students worked in small groups to estimate the shortest route between two cities, and then to compute this distance. No procedure was given, and groups explored various methods for finding the answer, including use of the calculator, diagrams, and mental calculation. Following the small group activity, students discussed these various methods used in the estimation and computation.
The purpose of this study is to compare fifthgraders in the UCSMP curriculum to the baseline data provided by Reys and her colleagues (1993). It is expected that students who have had the opportunity to explore various solution methods, including their own invented methods, should have developed a good number sense and flexibility in choosing operations, the very characteristics that are associated with good mental calculators. However, there are reasons to expect that these UCSMP students would not do as well on this test. Compared to traditional mathematics curricula, less time is allotted to practicing computation and more time is spent on geometry, data, number concepts, and problem solving. Consequently, teachers who use it have expressed concerns about their students' computational abilities, especially on standardized tests. And although the UCSMP curriculum provides opportunities for inventing and investigating various methods, powerful strategies such as use of compatible numbers or front-end addition (Hope, Reys, and Reys, 1987) are not specifically taught. Thus, student-invented algorithms that are efficient for 25 + 82 may not carry over to problems like 75 + 85 + 25 + 2000 if they do not sufficiently reduce the load on short-term memory.
Results from the whole-class test and an individual preference survey are reported. Following this, the processes involved in mental calculation are discussed along with implications for school instruction.
Method
Participants
Students in four fifth-grade classes using the fieldtest version of the UCSMP curriculum were tested using items from the whole-class computation test developed by Reys et al. (1993). Students at all schools had been using the UCSMP curriculum since kindergarten. The schools were selected so as to be similar to those in the Reys et al. study. Three of the classes were suburban schools and the fourth was a parochial school in a large city and all classes were of average ability.
Procedure
Procedures followed were identical to those used by Reys and her colleagues (for details see Reys, Reys, and Hope ,1993). Twenty-five items chosen from the Reys fifth-grade test were administered as a wholeclass test by a research assistant. The questions consisted of six addition, six subtraction, ten multiplication, and three division problems. Eleven of these problems were presented orally, ten problems were presented visually, and four story problems were presented in both modes. For the items presented orally, each was read twice-and students had about 8 seconds to record their answer. Visually presented items were presented on the overhead for the same amount of time but were not read aloud. Story problems were read and presented on the overhead. All problems were suited for a mental solution.
Students were instructed to solve the problems in their head, and a narrow piece of paper was provided with room for the answer but not for calculations. The tests were administered in April. Two classes (one urban parochial and one suburban public school) were also administered the Student Preference Survey used by Reys and her colleagues.
Several weeks after the whole-class test, five students in the urban parochial class who returned parent permission slips were individually interviewed as they solved ten problems mentally. Students were probed for their solution method and these interviews were videotaped and analyzed to clarify their solution methods.
Results
The UCSMP students showed a strong ability to calculate mentally. On the 25-item whole-class test, their overall mean score of 47% correct compared to 24% for the baseline group. On all but one of the questions, they outperformed students in the baseline group, and, for most questions, this difference was significant (see Table 1). For example, nine of the questions were answered correctly by twice as many of the UCSMP students proportionately as those in the baseline group. On seven of these questions more than three times as many did so. The greatest differences were on problems involving multiplication and division of powers of 10 (e.g., 60 x 70) and story problems of all types, although large differences were present across nearly all types of problems. Mean correct scores for the UCSMP classes ranged from 41% to 50%, and two-tailed t-tests showed no significant differences between the four classes.
Unlike the baseline sample, the UCSMP students did well on the division problems. For example, while only 12% of the baseline group correctly answered 3800 + 10, 72% of the UCSMP students did so. More generally, the UCSMP students did much better than the baseline group on all multiplication and division problems involving powers of ten, and unlike the baseline group, they scored higher on division problems than on many multiplication problems. These results probably reflect the link made between the operations in the UCSMP curriculum, e.g., a problem like 28 t 7 is often presented or discussed as 7 x _ = 28 rather than as a separate division fact. For this reason, division is not necessarily seen as simply a separate operation but as one related directly to multiplication. Additionally, estimation (e.g., 473 x 28 is about 500 x 30) is applied in a variety of contexts giving students experiences with multiplication and division using powers of tens.
UCSMP students also performed much higher than the baseline students on each of the story problems. In fact, UCSMP students did better on some applied problems than on similar symbolic problems. This is most likely in part a result of the common role of application in the curriculum, i.e., computation is nearly always practiced in some type of context rather than as a symbolic operation. From kindergarten onward, story problems play a major role in the curriculum, with students often constructing, sharing, and solving their own stories.
UCSMP students did better on addition problems that required chaining from left to right. For example, 38% of the UCSMP students solved 75 + 85 + 25 + 2000 compared to 1% of the baseline sample. Previous interviews with UCSMP students in second and fourth grade (Carroll & Balfanz, 1995) indicated that many had developed mental techniques that reduced the load on memory by adding left to right, such as "53 plus 79 is 120. Plus 9 is 129. Plus 3 is 132." In this manner, the number of addends is reduced quickly and efficiently. Students also used other efficient methods, such as counting up or down by hundreds and tens, or by decomposing numbers to simplify a problem.
The two problems on which the UCSMP students did the poorest were "265 minus 98" and "25 x 28." It was surprising that only 8% answered the first correctly since the problem could be easily solved as (265 - 100) + 2. However, because this problem was presented orally, this format may have increased the load on their memory, making them less likely to apply this strategy, i.e., attention might be focused on remembering 265 and 98, leaving little cognitive space with which to change 98 to 100 + 2. The low score on 25 x 28 reflects the fact students had not been taught specific mental strategies, e.g., 25 can be rewritten as 100/ 4. This is supported by their response on the student preference survey where only 3% of the UCSMP students chose mental computation as their preferred method for this problem, indicating that the students might benefit if certain less obvious strategies were introduced in the curriculum.
Student Preference Survey
Responses to the preference survey indicated two major differences between the UCSMP students and the students in Reys et al. baseline group. First, although the calculator was the least preferred method on all but one problem, the UCSMP students were proportionately nearly twice as likely to choose it compared to the baseline group. Second, UCSMP students seemed to be a bit more aware of what problems did not lend themselves to mental calculation. For example, 13% of the baseline group thought that 33 x 88 would best be solved mentally, while only 3% of the UCSMP group did so. Similarly, 29% of the baseline students indicated that mental computation would be their preferred method for solving 29 x 31 compared to 12% of the UCSMP students. These two results reflect the UCSMP students greater familiarity with both the use of calculators and mental procedures as alternatives to pencil-and-paper computation. They also indicate that as a result of their experiences in mathematics, the UCSMP students have better metacognitive skills in choosing efficient procedures during problem solving.
It should be noted that when the UCSMP students chose "pencil-and-paper" as the preferred method, this did not necessarily mean the standard written algorithms - which are never taught in the UCSMP curriculum. Instead, after having opportunities to work with their own methods of multiplication (such as repeated addition), students are introduced to some alternative methods for multiplication such as lattice multiplication and the use of partial products, e.g., 38 x 7 = (30 x 7) + (8 x 7), which lends itself to mental computation.
Student Interviews
Although no interviews were conducted in the Reys et al. study, five UCSMP students were individually interviewed and videotaped as they solved ten problems. Students interviewed had been in the UCSMP curriculum for at least two year, and all showed characteristics of good mental calculation and number sense (Sowder, 1992; Hope and Sherrill, 1987). For example, none of the students attempted to apply the standard written algorithms mentally, instead using procedures that reduced the numbers that they needed to remember. Students also showed flexibility, applying different mental procedures to problems involving the same operations. They also applied relationships between operations (e.g., subtraction as adding up) to simplify problems.
Table 1.
Previous research with primary students who are allowed to invent their own methods indicate a tendency to add and subtract from left to right, i.e., to start with the largest place value. UCSMP students experience with their own "invented algorithms" in the primary grades obviously carried over to their knowledge of mental calculation (Kamii, Lewis, and Livingston 1993) and were reflected in their solution methods. As one student explained during the interview, "It's easier for me to start with the bigger number, " that is, the digits with the largest place value. In solving 68 + 32, all students interviewed first added the 60 and the 30 and then the units. All talked of adding 60 and 30, rather than 6 and 3, indicating a good awareness of place value, and all answered this question correctly. Similarly, when subtracting $12.15 from $20, most found the difference between 20 and 12 first, and then readjusted the remaining cents in one step.
This preference for left to right was evident on many problems, including the division word problem (first dividing up the dollars and then the cents) and all multiplication problems. However, other strategies were also applied when they were more appropriate. For example, on the problem 426 + 75, students were more likely to make use of near compatible numbers, e.g.,26+75=75+25+ 1.
Student also used the relationship between operations to simplify problems. Students were more likely to solve "Double 84" as addition (80 + 80 + 4 + 4) rather than multiplication and to solve 100 - 65 as an addition missing-addend problem, 65 +_= 100. Students were also likely to make use of tens and hundreds to simplify problems. For example, in doubling 84, one student thought of it as 84 + 84, and then readjusted it as 80 + 20 + 60 + 8 = 100 + 68, and in solving 100 - 65, two students used a counting up to 100 by tens. Thus, in choosing an appropriate solution, students seemed to consider various aspects of the problem. This flexibility was also apparent in their explanations. During the follow-up probe, students would sometimes switch to another solution method or explain that they could have solved the problem by another method.
Overall, the interviews indicated that experiences in the primary grades exploring numbers (including place value), inventing and discussing various solution methods, and linking the operations were important building blocks in developing an ability to calculate mentally. Rather than approaching the problems algorithmically, i.e., as a subtraction or multiplication problem, students used their knowledge about number and operations to choose a method appropriate to the problem.
Discussion
Traditional mathematics instruction has typically ignored mental computation, as indicated by the low scores in the Reys et al. study (1993). These results are not surprising considering the focus of traditional curricula on paper-and-pencil algorithms. If mental computation is an ability we want students to develop, an analysis of the component skills and processes is needed as well as an understanding of what classroom activities best encourage mental computation. Some mental mathematics programs are presented as separate lessons in text or as brief warm-ups separate from the lesson, focusing on one strategy at a time. Although the strategies learned may be interesting to students, it is uncertain how effectively they will transfer and use these procedures during problem solving. It is even more doubtful that transfer will be achieved if testing and the bulk of mathematical activities focus on practicing written algorithms.
An alternative model for building mental computation in students is represented by curricula and instruction that allow for exploration of numbers and operations. Rather than as a series of skills, strategies, or mental tricks, knowledge of number develops within a larger context. In these curricula, primary grade students are encouraged over time to invent their own algorithms rather than practice school procedures, and fact knowledge develops around relationships (6 + 7 = 6 + 6 + 1 or 17 - 9 = 10 - 9 + 7). Students are encouraged to make reasonable estimates, communicate their mathematical reasoning, and use calculators and other tools. Process (reasoning and understanding) is given as much attention that the mathematical product, i.e., the answer. Mental computation is one way in which students do calculations, and along with estimation, it is integrated throughout the curriculum. It is important to note that except for transfer-ins, the UCSMP students in this student had these experiences since kindergarten, and thus the results here are longitudinal in nature.
This model is built upon the idea that mental computation is one component of an interconnected web of knowledge, sometimes called number sense. Rather than retrieving a particular algorithm to solve addition problems and another to solve subtraction problems, students who possess number sense should be more sensitive to the reasonableness of a solution method as well as the reasonableness of an answer. The results of this study, as well as additional research, supports this idea and illustrates the capabilities of students in mental computation when they have these experiences.
For example, in the study during the late seventies, Davis and McKnight (in Fuson, 1992) found that none of the above-average third and fourth graders could solve the problem 7000 - 2. The reason is most probably because students attempted to apply the standard written algorithm rather than observing the relative size of the number and accessing relevant knowledge of the number system,. The mathematical knowledge of these students was bound to an algorithm - in this case, one that was not well matched to the problem. In contrast, fourth-graders in UCSMP were interviewed on this same problem and two-thirds could correctly solve the problem, generally by counting down, by changing the problem ("I thought of 100 minus 2'), or by some similar method that required and exhibited number sense (Carroll and Porter, 1994).
Mental computation, as part of the larger domain of number sense, most probably develops best over an extended period of time in a variety of contexts and applications rather than as a separate strand taught in relative isolation. As a component of the domain, mental computation both is supported by and enhances number sense as well as metacognitive skills such as considering the most appropriate solution process. Rather than being algorithmic in nature, good mental computation is flexible and responsive to relative number size and operation. Some of the skills and processes observed during interviews with the students are shown in Table 2. Many of these, such as knowledge of invented procedures, stem directly from their experiences in the UCSMP primary curriculum.
However, not all of these processes or skills were used by students. For example, estimation and metacognition (monitoring the meaningfulness of the answer) were sometimes ignored, resulting in unreasonable answers. Nor was the choice of solution methods one that could be modeled as a hierarchy or a flow chart. Students recognized number or operation relationships that facilitated a solution on some problems but not on others. During follow-up explanations, students sometimes switched strategies, e.g., from initially chaining sums to using compatible numbers. This shows the complexity of helping students attain good number sense and, more specifically, skill at using mental computation wisely.
As a group, the students who had been in the reform-based curriculum showed a much better ability to compute mentally, and this has been supported by related research (Carroll and Balfanz, 1995; Carroll and Porter, 1994). Students were especially strong relative to the comparison group in areas emphasized inthe UCSMP curriculum, such as estimations that use of multiplication and division involving powers of tens. Still, the results indicate that there are gains to be made by this group, perhaps by including instruction in specific strategies that students may be less likely to "invent" or discover on their own. However it is unclear which strategies should be introduced, when this should occur is unclear, and whether they will be as effective as methods students develop on their own.
Table 2.
A number of research questions are raised by these results. How does mental computation and written computation interact? Standardized tests indicate that computation scores are lower than scores on concept and application subtests for UCSMP students. But these computation tests may be more likely to assess knowledge of written algorithms rather than a larger knowledge of number and operation. In the long run, how much does competence in mental computation add meaning to written and calculator procedures and how can this be best achieved? The relationship between estimation skills and mental computation was not assessed in this study and warrants investigation. Overall, these results show that students are much more capable of learning and using mental computation than has been expected of students in traditional curricula. Results also suggest that the components of number sense, such as estimation and number explorations, need to be on-going and integrated throughout instruction and learning.
Carroll, W. M. and Balfanz, R. (1995) Invented Algorithms of Second Graders in Reform Mathematics Curriculum. Paper presented at the 1995 Annual Meeting of the American Educational Research Association, San Francisco, CA.
Carroll, W. M. and Porter, D. (1994). A Field Test of Fourth Grade Everyday Mathematics. Chicago: University of Chicago School Mathematics Project.
Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23, 45-55.
Fuson, K C. (1992). Research on Whole Number Addition and Subtraction. In D. A. Grouws, (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 243-275). New York: Macmillan Publishing Company.
Hope, J. A. (1986). Mental calculation: Anachronism or
basic skill. In H. L. Schoen and M. J. Zweng (Eds.), Estimation and Mental Computation(pp. 45- 54).Reston, VA: National Council of Teachers of Mathematics. Hope, J. A., Reys, B. A., Reys, R E. (1987). Mental Math in the Middle Grades. Palo Alto, CA: Dale Seymour Publications.
Hope, J. A. and Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education,18, 98-111. Kamii, C., Lewis, B.A., & Livingston, S. J. (1993) Primary arithmetic: Children inventing their own procedures. Arithmetic Teacher, 41, 200 - 203. Reys, R. E., Bestgen, B. J., Rybolt, J. F. and Wyatt, J. W. (1982). Processes used by good estimators. Journal for Research in Mathematics Education, 13, 183-201. Reys, B. J., Reys, R. E., and Hope, J. A. (1993) Mental computation: A snapshot of second, fifth, and seventh grade student performance. School Science and Mathematics, 93, 306 - 315.
National Council of Teachers of Mathematics. (1989). The Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991) Professional Standards for Teaching Mathematics. Reston, VA: Author.
National Research Council. (1989). Everybody Counts. A Report on the Future of Mathematics Education. Washington, D.C.: National Academy Press.
Schoenfeld, A. H. (1992). Learning to think metacognitively: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws, (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370). New York: Macmillan Publishing Company.
Sowder, J. Estimation and number sense. In D. A. Grouws, (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 371-389-370). New York: Macmillan Publishing Company.
Wheatley, G. H., & Shumway, R., (1992). The potential for calculators to transform elementary school mathematics. In J. T. Fey & C. R. Hirsch, eds., Calculators in Mathematics Education: 1992 Yearbook (pp. 1 8). Reston, VA: The National Council of Teachers of Mathematics.
Author's Note: The research described in this report has been funded by the National Science Foundation Grant ESI-9252984.
Editor's Note: William M. Carroll's postal address is University of Chicago, Department of Education, 5835 South Kimbark Avenue, Chicago, IL 60637, and e-mail address is will@cicero.spc.uchicago.edu.
Abstract (Summary)
Fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems, and baseline data with students in a traditional mathematics curricula were used as a comparison. The students in the reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial.
» Jump to indexing (document details)
(4519 words)
When faced with a problem involving mathematical calculations, various solution methods are possible, including the use of paper-and-pencil algorithms, a calculator, or a mental solution. Although schools have generally emphasized standard written algorithms in both instruction and testing, the calculator has received an increasing acceptance as a regular tool from the primary grades onward (e.g., Wheatley & Shumway, 1992). In contrast, mental arithmetic is underrepresented in instruction and is rarely included on high-stakes tests. This is surprising, given that mental calculations and estimations are more common in everyday life than are the paper-and-pencil algorithms practiced in school (Hope, 1986). Current mathematics reform initiatives have recognized the need to align school and everyday uses of mathematics by including mental arithmetic and estimation as regular components in the curriculum (NCTM, 1989).
Besides better preparing students for the mathematics encountered outside of school, experiences with mental computation should help to develop flexible mathematical reasoning, estimation skills, alternative ways of expressing numbers, and the relationship between operations. Rather than focusing simply on the product, (i.e., the correct answer), mental computation allows greater opportunity to view mathematical problem solving as a process in a domain of interconnected knowledge sometimes called number sense (Sowder, 1992). For example, when adding 98 + 17, people with a good number sense may simplify the problem to (98 + 2) + IS or 100 + 20 - 5. They may also note that the sum will be about, but not quite, 120. Experiences with mental arithmetic also help students to recognize when a problem lends itself to a mental calculation and when it is better to use a calculator or pencil, metacognitive skills such as these are important in mathematical problem solving (Schoenfeld, 1992).
Perhaps most importantly, use of mental computation is motivating for many students. For example, in a recent survey (Carroll & Porter, 1994), fourth graders were asked to compare the mathematical knowledge of those who used a mental, paper-pencil, or calculator solution. Nearly all fourth-grade students surveyed felt that the student who used mental computation were better at mathematics.
Good mental calculators use procedures not necessarily learned in school (Dowker, 1992; Hope & Sherrill, 1987; Reys, Bestgen, Rybolt, & Wyatt, 1982) but which build upon number and operation relationships in an efficient manner. For example, good mental calculators often add from left-to-right, recombining numbers as they go along, thus reducing the load on memory. However, without frequent opportunities to develop mental solutions or modeling by the teacher or other students, many students do not develop these efficient techniques. Instead, they attempt to apply standard pencil-and-paper algorithms mentally (Hope & Sherrill, 1987). Due to the strain on short-term memory, these right-to-left calculations with "carrying" and "borrowing" are error-prone when used mentally.
A recent study of by Reys, Reys, and Hope (1993) attempted to set benchmarks in mental computation for comparative purposes. Results on this whole-class test, which tested classes using traditional mathematics curricula, highlighted weaknesses of students in mental computation. For example, only about one-third of fifth graders could mentally add 47 and 29 or could solve a story problem which required subtracting 65 from 100. Few problems were solved by more than half of the students. A preference survey indicated that students chose pencil-and-paper as their preferred method on most problems, even those that lent themselves to mental calculation. The calculator was the least preferred method on all items.
While traditional mathematics curricula emphasize practice of paper-and-pencil algorithms, a number of new mathematics curricula have attempted to incorporate ideas recommended by the current reform movement (NCTM, 1989, 1991; NRC, 1989). In these curricula, the development of estimation skills and number sense of answers is given more emphasis than paper-and-pencil computation. Primary students are often not taught specific algorithms but instead are encouraged to develop and share their own methods. Often these "invented" algorithms are mental and reflect methods used by expert mental calculators, e.g., adding left to right or decomposing numbers (Carroll & Balfanz, 1995; Porter & Carroll, 1995). One such curriculum in wide use is the University of Chicago School Mathematics Project elementary program, Everyday Mathematics. In addition to inventing their own algorithms, students in this curriculum are encouraged to share multiple solutions during group work and whole-class discussion. Manipulatives and calculators are used regularly during problem solving, and computation problems are nearly always presented in an applied situation or in a game. Mental arithmetic is emphasized in the curriculum, both implicitly by encouraging invented algorithms, and explicitly by encouraging the sharing of alternative strategies and short mental arithmetic exercises.
For example, in a typical lesson fourth-grade UCSMP lesson observed, students used with a road map showing various routes and mileage between cities. Students worked in small groups to estimate the shortest route between two cities, and then to compute this distance. No procedure was given, and groups explored various methods for finding the answer, including use of the calculator, diagrams, and mental calculation. Following the small group activity, students discussed these various methods used in the estimation and computation.
The purpose of this study is to compare fifthgraders in the UCSMP curriculum to the baseline data provided by Reys and her colleagues (1993). It is expected that students who have had the opportunity to explore various solution methods, including their own invented methods, should have developed a good number sense and flexibility in choosing operations, the very characteristics that are associated with good mental calculators. However, there are reasons to expect that these UCSMP students would not do as well on this test. Compared to traditional mathematics curricula, less time is allotted to practicing computation and more time is spent on geometry, data, number concepts, and problem solving. Consequently, teachers who use it have expressed concerns about their students' computational abilities, especially on standardized tests. And although the UCSMP curriculum provides opportunities for inventing and investigating various methods, powerful strategies such as use of compatible numbers or front-end addition (Hope, Reys, and Reys, 1987) are not specifically taught. Thus, student-invented algorithms that are efficient for 25 + 82 may not carry over to problems like 75 + 85 + 25 + 2000 if they do not sufficiently reduce the load on short-term memory.
Results from the whole-class test and an individual preference survey are reported. Following this, the processes involved in mental calculation are discussed along with implications for school instruction.
Method
Participants
Students in four fifth-grade classes using the fieldtest version of the UCSMP curriculum were tested using items from the whole-class computation test developed by Reys et al. (1993). Students at all schools had been using the UCSMP curriculum since kindergarten. The schools were selected so as to be similar to those in the Reys et al. study. Three of the classes were suburban schools and the fourth was a parochial school in a large city and all classes were of average ability.
Procedure
Procedures followed were identical to those used by Reys and her colleagues (for details see Reys, Reys, and Hope ,1993). Twenty-five items chosen from the Reys fifth-grade test were administered as a wholeclass test by a research assistant. The questions consisted of six addition, six subtraction, ten multiplication, and three division problems. Eleven of these problems were presented orally, ten problems were presented visually, and four story problems were presented in both modes. For the items presented orally, each was read twice-and students had about 8 seconds to record their answer. Visually presented items were presented on the overhead for the same amount of time but were not read aloud. Story problems were read and presented on the overhead. All problems were suited for a mental solution.
Students were instructed to solve the problems in their head, and a narrow piece of paper was provided with room for the answer but not for calculations. The tests were administered in April. Two classes (one urban parochial and one suburban public school) were also administered the Student Preference Survey used by Reys and her colleagues.
Several weeks after the whole-class test, five students in the urban parochial class who returned parent permission slips were individually interviewed as they solved ten problems mentally. Students were probed for their solution method and these interviews were videotaped and analyzed to clarify their solution methods.
Results
The UCSMP students showed a strong ability to calculate mentally. On the 25-item whole-class test, their overall mean score of 47% correct compared to 24% for the baseline group. On all but one of the questions, they outperformed students in the baseline group, and, for most questions, this difference was significant (see Table 1). For example, nine of the questions were answered correctly by twice as many of the UCSMP students proportionately as those in the baseline group. On seven of these questions more than three times as many did so. The greatest differences were on problems involving multiplication and division of powers of 10 (e.g., 60 x 70) and story problems of all types, although large differences were present across nearly all types of problems. Mean correct scores for the UCSMP classes ranged from 41% to 50%, and two-tailed t-tests showed no significant differences between the four classes.
Unlike the baseline sample, the UCSMP students did well on the division problems. For example, while only 12% of the baseline group correctly answered 3800 + 10, 72% of the UCSMP students did so. More generally, the UCSMP students did much better than the baseline group on all multiplication and division problems involving powers of ten, and unlike the baseline group, they scored higher on division problems than on many multiplication problems. These results probably reflect the link made between the operations in the UCSMP curriculum, e.g., a problem like 28 t 7 is often presented or discussed as 7 x _ = 28 rather than as a separate division fact. For this reason, division is not necessarily seen as simply a separate operation but as one related directly to multiplication. Additionally, estimation (e.g., 473 x 28 is about 500 x 30) is applied in a variety of contexts giving students experiences with multiplication and division using powers of tens.
UCSMP students also performed much higher than the baseline students on each of the story problems. In fact, UCSMP students did better on some applied problems than on similar symbolic problems. This is most likely in part a result of the common role of application in the curriculum, i.e., computation is nearly always practiced in some type of context rather than as a symbolic operation. From kindergarten onward, story problems play a major role in the curriculum, with students often constructing, sharing, and solving their own stories.
UCSMP students did better on addition problems that required chaining from left to right. For example, 38% of the UCSMP students solved 75 + 85 + 25 + 2000 compared to 1% of the baseline sample. Previous interviews with UCSMP students in second and fourth grade (Carroll & Balfanz, 1995) indicated that many had developed mental techniques that reduced the load on memory by adding left to right, such as "53 plus 79 is 120. Plus 9 is 129. Plus 3 is 132." In this manner, the number of addends is reduced quickly and efficiently. Students also used other efficient methods, such as counting up or down by hundreds and tens, or by decomposing numbers to simplify a problem.
The two problems on which the UCSMP students did the poorest were "265 minus 98" and "25 x 28." It was surprising that only 8% answered the first correctly since the problem could be easily solved as (265 - 100) + 2. However, because this problem was presented orally, this format may have increased the load on their memory, making them less likely to apply this strategy, i.e., attention might be focused on remembering 265 and 98, leaving little cognitive space with which to change 98 to 100 + 2. The low score on 25 x 28 reflects the fact students had not been taught specific mental strategies, e.g., 25 can be rewritten as 100/ 4. This is supported by their response on the student preference survey where only 3% of the UCSMP students chose mental computation as their preferred method for this problem, indicating that the students might benefit if certain less obvious strategies were introduced in the curriculum.
Student Preference Survey
Responses to the preference survey indicated two major differences between the UCSMP students and the students in Reys et al. baseline group. First, although the calculator was the least preferred method on all but one problem, the UCSMP students were proportionately nearly twice as likely to choose it compared to the baseline group. Second, UCSMP students seemed to be a bit more aware of what problems did not lend themselves to mental calculation. For example, 13% of the baseline group thought that 33 x 88 would best be solved mentally, while only 3% of the UCSMP group did so. Similarly, 29% of the baseline students indicated that mental computation would be their preferred method for solving 29 x 31 compared to 12% of the UCSMP students. These two results reflect the UCSMP students greater familiarity with both the use of calculators and mental procedures as alternatives to pencil-and-paper computation. They also indicate that as a result of their experiences in mathematics, the UCSMP students have better metacognitive skills in choosing efficient procedures during problem solving.
It should be noted that when the UCSMP students chose "pencil-and-paper" as the preferred method, this did not necessarily mean the standard written algorithms - which are never taught in the UCSMP curriculum. Instead, after having opportunities to work with their own methods of multiplication (such as repeated addition), students are introduced to some alternative methods for multiplication such as lattice multiplication and the use of partial products, e.g., 38 x 7 = (30 x 7) + (8 x 7), which lends itself to mental computation.
Student Interviews
Although no interviews were conducted in the Reys et al. study, five UCSMP students were individually interviewed and videotaped as they solved ten problems. Students interviewed had been in the UCSMP curriculum for at least two year, and all showed characteristics of good mental calculation and number sense (Sowder, 1992; Hope and Sherrill, 1987). For example, none of the students attempted to apply the standard written algorithms mentally, instead using procedures that reduced the numbers that they needed to remember. Students also showed flexibility, applying different mental procedures to problems involving the same operations. They also applied relationships between operations (e.g., subtraction as adding up) to simplify problems.
Previous research with primary students who are allowed to invent their own methods indicate a tendency to add and subtract from left to right, i.e., to start with the largest place value. UCSMP students experience with their own "invented algorithms" in the primary grades obviously carried over to their knowledge of mental calculation (Kamii, Lewis, and Livingston 1993) and were reflected in their solution methods. As one student explained during the interview, "It's easier for me to start with the bigger number, " that is, the digits with the largest place value. In solving 68 + 32, all students interviewed first added the 60 and the 30 and then the units. All talked of adding 60 and 30, rather than 6 and 3, indicating a good awareness of place value, and all answered this question correctly. Similarly, when subtracting $12.15 from $20, most found the difference between 20 and 12 first, and then readjusted the remaining cents in one step.
This preference for left to right was evident on many problems, including the division word problem (first dividing up the dollars and then the cents) and all multiplication problems. However, other strategies were also applied when they were more appropriate. For example, on the problem 426 + 75, students were more likely to make use of near compatible numbers, e.g.,26+75=75+25+ 1.
Student also used the relationship between operations to simplify problems. Students were more likely to solve "Double 84" as addition (80 + 80 + 4 + 4) rather than multiplication and to solve 100 - 65 as an addition missing-addend problem, 65 +_= 100. Students were also likely to make use of tens and hundreds to simplify problems. For example, in doubling 84, one student thought of it as 84 + 84, and then readjusted it as 80 + 20 + 60 + 8 = 100 + 68, and in solving 100 - 65, two students used a counting up to 100 by tens. Thus, in choosing an appropriate solution, students seemed to consider various aspects of the problem. This flexibility was also apparent in their explanations. During the follow-up probe, students would sometimes switch to another solution method or explain that they could have solved the problem by another method.
Overall, the interviews indicated that experiences in the primary grades exploring numbers (including place value), inventing and discussing various solution methods, and linking the operations were important building blocks in developing an ability to calculate mentally. Rather than approaching the problems algorithmically, i.e., as a subtraction or multiplication problem, students used their knowledge about number and operations to choose a method appropriate to the problem.
Discussion
Traditional mathematics instruction has typically ignored mental computation, as indicated by the low scores in the Reys et al. study (1993). These results are not surprising considering the focus of traditional curricula on paper-and-pencil algorithms. If mental computation is an ability we want students to develop, an analysis of the component skills and processes is needed as well as an understanding of what classroom activities best encourage mental computation. Some mental mathematics programs are presented as separate lessons in text or as brief warm-ups separate from the lesson, focusing on one strategy at a time. Although the strategies learned may be interesting to students, it is uncertain how effectively they will transfer and use these procedures during problem solving. It is even more doubtful that transfer will be achieved if testing and the bulk of mathematical activities focus on practicing written algorithms.
An alternative model for building mental computation in students is represented by curricula and instruction that allow for exploration of numbers and operations. Rather than as a series of skills, strategies, or mental tricks, knowledge of number develops within a larger context. In these curricula, primary grade students are encouraged over time to invent their own algorithms rather than practice school procedures, and fact knowledge develops around relationships (6 + 7 = 6 + 6 + 1 or 17 - 9 = 10 - 9 + 7). Students are encouraged to make reasonable estimates, communicate their mathematical reasoning, and use calculators and other tools. Process (reasoning and understanding) is given as much attention that the mathematical product, i.e., the answer. Mental computation is one way in which students do calculations, and along with estimation, it is integrated throughout the curriculum. It is important to note that except for transfer-ins, the UCSMP students in this student had these experiences since kindergarten, and thus the results here are longitudinal in nature.
This model is built upon the idea that mental computation is one component of an interconnected web of knowledge, sometimes called number sense. Rather than retrieving a particular algorithm to solve addition problems and another to solve subtraction problems, students who possess number sense should be more sensitive to the reasonableness of a solution method as well as the reasonableness of an answer. The results of this study, as well as additional research, supports this idea and illustrates the capabilities of students in mental computation when they have these experiences.
For example, in the study during the late seventies, Davis and McKnight (in Fuson, 1992) found that none of the above-average third and fourth graders could solve the problem 7000 - 2. The reason is most probably because students attempted to apply the standard written algorithm rather than observing the relative size of the number and accessing relevant knowledge of the number system,. The mathematical knowledge of these students was bound to an algorithm - in this case, one that was not well matched to the problem. In contrast, fourth-graders in UCSMP were interviewed on this same problem and two-thirds could correctly solve the problem, generally by counting down, by changing the problem ("I thought of 100 minus 2'), or by some similar method that required and exhibited number sense (Carroll and Porter, 1994).
Mental computation, as part of the larger domain of number sense, most probably develops best over an extended period of time in a variety of contexts and applications rather than as a separate strand taught in relative isolation. As a component of the domain, mental computation both is supported by and enhances number sense as well as metacognitive skills such as considering the most appropriate solution process. Rather than being algorithmic in nature, good mental computation is flexible and responsive to relative number size and operation. Some of the skills and processes observed during interviews with the students are shown in Table 2. Many of these, such as knowledge of invented procedures, stem directly from their experiences in the UCSMP primary curriculum.
However, not all of these processes or skills were used by students. For example, estimation and metacognition (monitoring the meaningfulness of the answer) were sometimes ignored, resulting in unreasonable answers. Nor was the choice of solution methods one that could be modeled as a hierarchy or a flow chart. Students recognized number or operation relationships that facilitated a solution on some problems but not on others. During follow-up explanations, students sometimes switched strategies, e.g., from initially chaining sums to using compatible numbers. This shows the complexity of helping students attain good number sense and, more specifically, skill at using mental computation wisely.
As a group, the students who had been in the reform-based curriculum showed a much better ability to compute mentally, and this has been supported by related research (Carroll and Balfanz, 1995; Carroll and Porter, 1994). Students were especially strong relative to the comparison group in areas emphasized inthe UCSMP curriculum, such as estimations that use of multiplication and division involving powers of tens. Still, the results indicate that there are gains to be made by this group, perhaps by including instruction in specific strategies that students may be less likely to "invent" or discover on their own. However it is unclear which strategies should be introduced, when this should occur is unclear, and whether they will be as effective as methods students develop on their own.
A number of research questions are raised by these results. How does mental computation and written computation interact? Standardized tests indicate that computation scores are lower than scores on concept and application subtests for UCSMP students. But these computation tests may be more likely to assess knowledge of written algorithms rather than a larger knowledge of number and operation. In the long run, how much does competence in mental computation add meaning to written and calculator procedures and how can this be best achieved? The relationship between estimation skills and mental computation was not assessed in this study and warrants investigation. Overall, these results show that students are much more capable of learning and using mental computation than has been expected of students in traditional curricula. Results also suggest that the components of number sense, such as estimation and number explorations, need to be on-going and integrated throughout instruction and learning.
References
Indexing (document details)